For questions about paracompact spaces and partitions of unity, as well as variants such as metacompact spaces

A topological space is called paracompact if every open cover has a locally finite refinement. Some authors also require paracompact spaces to be Hausdorff.

Most applications of paracompactness arise from the fact that a paracompact Hausdorff space admits a partition of unity subordinate to any open cover. Partitions of unity are useful for "gluing" local structure on a space to obtain a global structure. Manifolds are commonly assumed to be paracompact so that they have partitions of unity.

This tag can also be used for questions on variants of paracompactness, such as metacompactness (every open cover has a point-finite refinement) or countable paracompactness (every countable open cover has a locally finite refinement).